Optimal error estimates for Nedelec edge elements for time-harmonic Maxwell's equations

Liuqiang Zhong, Shi Shu, Gabriel Wittum, Jinchao Xu

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

In this paper, we obtain optimal error estimates in both L2-norm and H(curl)-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L2 error estimates into the L2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nédélec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

Original languageEnglish (US)
Pages (from-to)563-572
Number of pages10
JournalJournal of Computational Mathematics
Volume27
Issue number5
DOIs
StatePublished - Sep 1 2009

Fingerprint

Edge Elements
Divergence-free
Optimal Error Estimates
Maxwell equations
Maxwell's equations
Harmonic
Edge Finite Elements
Norm
Convergence Estimates
Lipschitz Domains
Curl
Finite Element Approximation
Error Estimates
Duality
Mesh
Transform
Approximation
Estimate

All Science Journal Classification (ASJC) codes

  • Computational Mathematics

Cite this

@article{19de01b61cb244e8a2eaa368c44a3fc4,
title = "Optimal error estimates for Nedelec edge elements for time-harmonic Maxwell's equations",
abstract = "In this paper, we obtain optimal error estimates in both L2-norm and H(curl)-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L2 error estimates into the L2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For N{\'e}d{\'e}lec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.",
author = "Liuqiang Zhong and Shi Shu and Gabriel Wittum and Jinchao Xu",
year = "2009",
month = "9",
day = "1",
doi = "10.4208/jcm.2009.27.5.011",
language = "English (US)",
volume = "27",
pages = "563--572",
journal = "Journal of Computational Mathematics",
issn = "0254-9409",
publisher = "Inst. of Computational Mathematics and Sc./Eng. Computing",
number = "5",

}

Optimal error estimates for Nedelec edge elements for time-harmonic Maxwell's equations. / Zhong, Liuqiang; Shu, Shi; Wittum, Gabriel; Xu, Jinchao.

In: Journal of Computational Mathematics, Vol. 27, No. 5, 01.09.2009, p. 563-572.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Optimal error estimates for Nedelec edge elements for time-harmonic Maxwell's equations

AU - Zhong, Liuqiang

AU - Shu, Shi

AU - Wittum, Gabriel

AU - Xu, Jinchao

PY - 2009/9/1

Y1 - 2009/9/1

N2 - In this paper, we obtain optimal error estimates in both L2-norm and H(curl)-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L2 error estimates into the L2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nédélec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

AB - In this paper, we obtain optimal error estimates in both L2-norm and H(curl)-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L2 error estimates into the L2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nédélec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

UR - http://www.scopus.com/inward/record.url?scp=70249100276&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70249100276&partnerID=8YFLogxK

U2 - 10.4208/jcm.2009.27.5.011

DO - 10.4208/jcm.2009.27.5.011

M3 - Article

AN - SCOPUS:70249100276

VL - 27

SP - 563

EP - 572

JO - Journal of Computational Mathematics

JF - Journal of Computational Mathematics

SN - 0254-9409

IS - 5

ER -